Group

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Group
A group is defined as a finite or infinite set of Operands
(called ``elements'') , , , ... that may be combined or ``multiplied'' via a Binary Operator
to form well-defined products and which furthermore satisfy the following conditions:
0 a) D7 ^5 F, I  g) C1. Closure: If and are two elements in , then the product is also in .
2. Associativity: The defined multiplication is associative, i.e., for all , .
3. Identity: There is an Identity Element
5 U2 @) D) \( q  Q (a.k.a. , , or ) such that for every element .
4. Inverse: There must be an inverse or reciprocal of each element. Therefore, the set must contain an element such that for each element of .
! v9 g% c1 t- q2 [A group is therefore a Monoid
for which every element is invertible. A group must contain at least one element.

4 i' J: F0 J7 _# a% W$ z, ^! hThe study of groups is known as Group Theory
3 t6 h9 g0 q+ y. p% p. If there are a finite number of elements, the group is called a Finite Group
0 l& c3 ]' Z! h* A7 p( ` and the number of elements is called the Order
of the group.

8 a9 F+ a  h; s. M3 y- ~6 DSince each element , , , ..., , and is a member of the group, group property 1 requires that the product
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(1)

0 z8 P6 p* a, C+ q) Omust also be a member. Now apply to ,
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+ Q) T7 k" Y: p

(2) 9 t" \* r! [7 a$ z3 `* ]" Y

; \5 y$ v2 A, X% f1 V0 U$ L. OBut
9 u& Y5 V% p: o

	) k& S" F" }3 @. g% {
: }0 j* F- E* R: h	(3)

so
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5 z2 O4 Z! ^4 _5 s8 T

(4)

6 s* o1 ~1 J* K0 b4 `
5 S9 z" I/ {* C1 C/ R" [. _  n
which means that

(5) ' J7 f: h- z0 j+ R) ]

2 L# d: B& T; J2 x5 d+ U2 ^
and

(6)

) e! d1 ^  e1 c& f& f
7 |  S9 H9 K1 y


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